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3. Algebraic Expressions
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Pre-Algebra /
Chapter 4
3. 

Algebraic Expressions

This lesson provides a comprehensive guide to understanding algebraic expressions, a cornerstone in the field of algebra. It introduces you to the basic elements like variables, which represent unknown quantities, and constants, which are fixed numbers. The lesson also explains the importance of coefficients, the numbers that multiply variables. Through real-world examples, such as calculating income or game scores, it shows how these elements are combined through mathematical operations like addition and subtraction to form algebraic expressions. The aim is to equip you with the skills to write and manipulate these expressions, making it easier to solve problems in academics and everyday life.
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16 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Algebraic Expressions
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Numerical expressions deal with combinations of mathematical operations with numbers, which are known quantities. However, sometimes it is necessary to deal with unknown quantities. This lesson will examine how to incorporate unknown quantities in expressions and how to evaluate that expression once the quantity is known.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Garage Sale to Buy a New Game

Tearrik is really excited about a game that is coming out this weekend. He decides to sell some of his stuff so that he can make enough money to buy the game.

A garage sale sign hangs on the bottom right, while a wooden table on the left displays a video game controller, a toy train, and clothing.
Proud of him for making efforts to buy his own things, Tearrik's parents decided to double the money he makes and add $5 extra on top. How can this situation be written in a mathematical way without knowing how much money Tearrik makes from the garage sale?
Discussion

Variable

A variable is a symbol used to represent an unknown quantity. Often, variables represent fixed but unknown numbers. Variables are usually denoted with letters such as x. x+1=8

Alternatively, a variable can be used to represent a quantity that changes. Izabella's income varies depending on the number of hours she works. She receives a fixed salary of$100 per week plus$4 per hour. In this case, Izabella's weekly income could be different every week, depending on the number of hours she works. Therefore, the use of a variable is appropriate. Let x be the hours that Izabella works in a week. Her income can be written by adding the fixed $100 to the $4 per hour.

Izabella's Weekly Income 4x+100
Discussion

The Numbers Next to Variables

When dealing with variables, sometimes other numbers are needed to complete a mathematical idea. One type of these numbers is coefficients.

Theory

Coefficient

A coefficient is a number that stands in front of a variable.

A coefficient always multiplies a variable. Consider the following example. 5x In this case, the coefficient is 5. The coefficient is the multiplier, even if the variable is raised to some power. 2x^3 In this case, the coefficient is 2. Note that a variable with a coefficient of 1 is usually written without a coefficient.

1x^2 = x^2
Discussion

Adding Numbers to a Variable

Another type of numbers that appear with variables is called constants.

Concept

Constant

A constant is a number with a known, fixed value. Examples of constants include regular numbers written with digits, like 15, - 23, and 11.327.

A constant is usually added to or subtracted from a variable. Consider the following example. x + 15

In this case, 15 is a constant. It should be noted that not every constant is written with digits. Some special constants are written with special symbols, such as the number pi, which is often written as π .
Pop Quiz

Practice with Variables and Coefficients

The applet below displays different variables multiplied by coefficients. Answer the indicated question correctly.

Variable or Coefficient?
Discussion

Algebraic Expression

An algebraic expression is a valid combination of numbers, variables, and mathematical operations. For example, in the expression 2x+3, the variable x is being multiplied by its coefficient 2, and this product is then added to the constant 3.

The figure illustrating the components of an algebraic expression, such as 2x+3. In this representation, 2 is the coefficient, x is the variable, and 3 is the constant term.
Discussion

Components of an Algebraic Expression

Algebraic expressions are made by adding or subtracting smaller expressions called terms.

Concept

Term - Expression

A term is an algebraic or numeric expression that does not involve addition or subtraction. An expression contains one or more terms, separated from one another by + or - signs.
Mathematical Expression Number of Terms Terms
7x 1 7x
8 1 8
8x - 2(5) 2 8x and -2(5)
x^2 + y^2 +4 3 x^2, y^2, and 4
2x^2 -5x - 122 3 2x^2, -5x, and -122
Example

The Cost of Clothes

At his garage sale, Tearrik is selling some of his old shirts and pants.

Tearrik showing a shirt and a pair of pants
He is selling the pants for $1 more than three times the price of a shirt. If the price of a shirt is s, write an algebraic expression for the price of a pair of pants.

Hint

Remember that an algebraic expression is a combination of numbers, variables, and mathematical operations.

Solution

An algebraic expression is a combination of numbers, variables, and mathematical operations. The terms are separated by + or - signs. In verbal expressions, some words or phrases may imply certain math operations.

Key Words and Phrases
Addition added to, plus, sum of, more than, increased by, total of and
Subtraction subtracted from, minus, difference of, less than, decreased by, fewer than, take away
Multiplication multiplied by, times, product of, twice
Division divided by, quotient of

It is given that the price of a shirt is written with the variable s. The operations will be found by examining the given information. Tearrik is selling the pants at $1 more than three times the price of a shirt. It is time to identify the keywords in this sentence. The phrase more than indicates an addition. In this case, $1 is going to be added to some other quantity. 1 + The phrase three times represents a multiplication. In this case, it is 3 times the price of a shirt, which is s. 1 + 3s There is no more information to include, so this expression represents the price of a pair of pants. 1+3s

Example

Bonus Points for a Challenge

Tearrik was able to buy the video game he desired and he started playing right away.

Video-game-level.jpg

In the game, timed challenges reward players with bonus points for finishing them quickly. Confused about how the bonus points are rewarded, Tearrik asked his friend Magdalena about it. She told him how the bonus points work.

The bonus points are half the difference between 400 and the time it took to fininsh the challenge
Help Tearrik write the information that Magdalena gave him as an algebraic expression. Let the variable be t.

Hint

Identify the variable. Then, look for keywords in Magdalena's information that indicate operations.

Solution

It is important to identify the variables before writing an algebraic expression. In Magdalena's description, the time it takes to finish the challenge can be different for different tries or different people. Since this time can change, it can be assigned to the given variable t. Time to Finish the Challenge: t An algebraic expression is a combination of numbers, variables, and mathematical operations. The terms are separated by + or - signs. In verbal expressions, there are words or phrases that indicate certain mathematical operations.

Key Words and Phrases
Addition added to, plus, sum of, more than, increased by, total of and
Subtraction subtracted from, minus, difference of, less than, decreased by, fewer than, take away
Multiplication multiplied by, times, product of, twice
Division divided by, quotient of

Examining Magdalena's explanation, it is possible to find some of these keywords. The bonus points are half the difference between400 and the time it took to finish the challenge The half indicates a division by 2 and the difference indicates a subtraction. Magdalena says to take half the difference. Because of this, it is convenient to write the difference first. This difference can be written by subtracting the time t from 400. 400 - t The division by 2 affects the result of the subtraction. Considering the order of operations, division operations are evaluated before subtraction. To evaluate the subtraction first, it is important to write the subtraction between parentheses. 1/2(400 - t) This expression can help Tearrik to determine how many bonus points he will get for however much time he takes to solve the challenge.

Discussion

Evaluating an Algebraic Expression

Evaluating an expression consists of determining the value of an expression when the variable or variables of the expression take a specific value. This is done by substituting the given value for the variable in question into the expression and then simplifying it. As an example, consider the following expression. (x-1)^2/2 This expression is going to be evaluated considering that x is equal to 5. There are two steps to follow.
1
Substitute the Given Value for x
expand_more
In the expression, substitute the given value for every instance of the variable. In this case, substitute 5 for x.
(x-1)^2/2
Substitute

x= 5

( 5-1)^2/2
After the substitution, the variable disappeared and the algebraic expression turned into a numeric expression.
2
Simplify the Expression
expand_more
Now the expression can be simplified following the order of operations.
(5-1)^2/2
SubTerms

Subtract terms

4^2/2
CalcPow

Calculate power

16/2
CalcQuot

Calculate quotient

8
Consequently, when x is equal to 5, the given expression equals 8.
If an expression has more than one variable, each variable is replaced by its given value before all the required operations are performed.
Example

Bonus Stars on Each Level

Tearrik continues to enjoy playing his new game. He is focusing on collecting the challenge stars on each level.

Video-game-star.jpg

a Tearrik wants to collect all the stars in the game. There are 3 special stars per level. If x is the number of levels, write an expression for the numbers of stars.
b There are 73 levels in the game. Use the expression from Part A to find the total number of stars.

Hint
a The number of stars per level indicates a multiplication.
b Substitute the given value for the variable and evaluate the expression.

Solution
a The total number of bonus stars can be found by adding the number of stars on every level. Every level has 3 stars. Adding the number of stars of every level is the same as multiplying 3 by the number of levels, x.

3+ 3+...+ 3_x= 3x

b There are 73 levels in the game, meaning that x is 73.
x = 73 Use this fact and evaluate the algebraic expression from Part A to find the total number of stars in the game. Notice that substituting the value for the variable changes the algebraic expression into a numerical expression.
3x
Substitute

x= 73

3* 73
Multiply

Multiply

219
There are 219 bonus stars in the game.
Example

The Difference Between the Area and the Perimeter of a Square

After a weekend of playing his new game, Tearrik has to go to class. During math class, the teacher drew a square on the whiteboard.

Example square with a side s

The professor asked the class about the difference between the area of the square and its perimeter.

a Help Tearrik to write an algebraic expression for the difference between the area of the square and its perimeter.
b If the side length of a square is 7, find the difference between its area and its perimeter.

Hint
a The area of a square is the square of the length of a side. The perimeter of a square is the sum of the lengths of all four sides.
b Use the algebraic expression written in Part A. Evaluate the expression when s=7.

Solution
a Start by recalling the formulas for the area and the perimeter of a square. The area of a square is given by the square of a side. The square that Tearrik's teacher drew indicates that each side has a length of s.

Area of the Square: s^2 The perimeter, on the other hand, is found by adding the length of all sides of a square. Since the four sides of the square all have a length of s, the perimeter is found by multiplying s by 4. Perimeter of the Square: 4s Then, to find the difference between these two values, subtract the perimeter from the area. s^2- 4s

b The teacher said that the side length of the square is 7. This means that s equals 7. Then, to find the difference between the area and the perimeter, evaluate the expression for the difference when s= 7.
s^2-4s
Substitute

s= 7

7^2 - 4* 7
CalcPow

Calculate power

49 - 4*7
Multiply

Multiply

49 - 28
SubTerms

Subtract terms

21
Therefore, the difference between the area and the perimeter of the given square is 21.


Example

Selling Shirts and Pants

While studying math, Tearrik remembered that he drew a sign to show the prices for the shirts and the pants during his garage sale.

Tearrik Sign with Shirts for 5 dollars and Pants for 16 dollars

Since Tearrik started studying algebraic expressions, he realized that he could use variables to represent the number of items of clothing he sold. He assigned s for the shirts he sold and p for the pants he sold.

a Help Tearrik write an algebraic expression for the total amount of money he made from selling shirts and pants.
b If Tearrik sold 10 shirts and 4 pairs of pants, how much did he make?

Hint
a Determine how much Tearrik would make from each type of clothing. Then, add these amounts to find the total.
b Substitute the appropriate values for the variables in the expression. Then, evaluate.

Solution
a Tearrik made money from selling pants and shirts. In other words, the total amount of money he made is the sum of what he got from selling shirts and pants.

Money From Shirts

Since every shirt costs $ 5 and he sold s shirts, the amount of money Tearrik made from them can be written as the product of 5 and s. 5* s

Money From Pants

In a similar way, the money made from selling pairs of pants is the product of their price, $ 16, and the number of pairs of pants sold, p. 16* p

Total Money Made

Finally, the expression for the total amount of money made comes from adding the two previous expressions. 5 s + 16 p

b The amount of money that Tearrik made from selling 10 shirts and 4 pairs of pants can be found by evaluating the expression.
Evaluate 5 s +16 p ⇓ when s = 10 and s = 4 Substitute the values for the corresponding variables and find the value of the resulting numerical expression.
5s + 16p
SubstituteII

s= 10, p= 4

5* 10 + 16 * 4
MultII

Multiply 5 by 10

50 + 16 * 4
MultII

Multiply 16 by 4

50 + 64
AddTerms

Add terms

114
Tearrik made a total of $114. It sounds like he had a very productive sale!


Pop Quiz

Practice Evaluating Algebraic Expressions

Consider the following variables with assigned values. a = 5 & b = 1/2 [0.8em] c = 2 & d = 14 Evaluate the given algebraic expression for the given values.

Numerical Expressions to Evaluate
Closure

Money Made From the Garage Sale

At the beginning of the lesson, Tearrik's parents decided to give him money based on what he made from selling his old clothes. How much money Tearrik made initially is unknown, but a variable can be used to represent this value. Consider x as the money Tearrik made at his sale. Money From the Garage Sale: x Consider what Tearrik's parents said about how much money they would give him. Keep in mind that phrases in this plan will indicate the necessary operations for writing the information as an algebraic expression. Double the money Tearrik made and add an additional $5  extra. The word double indicates multiplication by 2. This means that x is multiplied by 2. 2x On the other hand, the word add indicates addition. The $ 5 are added to double the money Tearrik made. 2x + 5

And now an expression for the total amount of money Tearrik got from his garage sale can be written, even if the initial amount is unknown. Algebraic expressions are great for writing mathematical ideas easily.




Level 1
Level 2
Level 3
Algebraic Expressions
Exercise 3.1
The difference between two numbers is 11. If the lesser number is x and the greater number is y, write an expression that represents the greater number in terms of x.

Let's look for keywords to write an expression for the greater number. The word difference indicates a subtraction. Since the variables x and y are used to represent the lesser and greater number, we can start writing the expression. y - x We are told that this difference is 11. We can write this by putting an equality symbol between y-x and 11. y - x = 11 Now we have an expression for the difference. This is great progress, but we want an expression for the number y. It is a good thing that when we add and subtract the same number, the result is always zero. We should keep in mind that we must add x to both sides of the equality symbol.

Note that we modified both sides of the equality symbol doing the same operation. When we do this, the equality remains true! Now we found an expression for the greater number y. y = 11 + x Therefore, the greater number is represented by the expression 11+x.

E
Hint & Answer
S
Solution

The area of a square is equal to 2 times the perimeter of the square. The dimensions of the square are in meters.

Square with side s
What is the area of the square?

Let's recall the formula for the area of a square. The area of a square is the square of the side's length. Since the given square has a side length of s, its area is the square of s. Area of the Square: s^2 We are told that this area is 2 times the perimeter of the square. Since the perimeter of a square is 4 times its side length, our square has a perimeter of 4 times s. Perimeter of the Square: 4s Now we can relate the area and perimeter of the square. The area of the square is equal to 2 times the perimeter of the square. s^2 & = 2 * 4s [0.4em] s^2 & = 8s Here, we can rewrite s^2 as s* s. Let's do it!. s* s = 8s We can see that the variable s is on both sides of the equality symbol. If we divide both sides by s, we can isolate s on one side of the equation.

The square has a length of 8 meters, but we need the area of the square. It is a good thing that we already know the expression for the area of the square. We substitute s=8 into s^2.

We concluded that the square has an area of 64 square meters.

E
Hint & Answer
S
Solution

Algebraic Expressions

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